Answer
$S$ spans $R^2$.
Work Step by Step
Assume the combination
$$a(1,-1)+b(2,1)=(0,0).$$
We have the system
\begin{align*}
a+2b&=0\\
-a+b&=0.
\end{align*}
Since the determinant of the matrix is given by
$$\left| \begin{array} {cc} 1&2\\-1&1 \end{array} \right|=3\neq 0$$
then the system has unique solution, that is, $$a=0, \quad b=0.$$
Consequently, $S$ is linearly independent and since $R^2$ has the dimension $2$ then $S$ spans $R^2$.
